12th AIME 1994

1. The sequence 3, 15, 24, 48, ... is those multiples of 3 which are one less than a square. Find the remainder when the 1994th term is divided by 1000.

2. The large circle has diameter 40 and the small circle diameter 10. They touch at P. PQ is a diameter of the small circle. ABCD is a square touching the small circle at Q. Find AB.

3. The function f satisfies f(x) + f(x-1) = x² for all x. If f(19) = 94, find the remainder when f(94) is divided by 1000.

4. Find n such that [log₂1] + [log₂2] + [log₂3] + ... + [log₂n] = 1994.

5. What is the largest prime factor of p(1) + p(2) + ... + p(999), where p(n) is the product of the non-zero digits of n?

6. How many equilateral triangles of side 2/√3 are formed by the lines y = k, y = x√3 + 2k, y = -x√3 + 2k for k = -10, -9, ... , 9, 10?

7. For how many ordered pairs (a, b) do the equations ax + by = 1, x² + y² = 50 have (1) at least one solution, and (2) all solutions integral?

8. Find ab if (0, 0), (a, 11), (b, 37) is an equilateral triangle.

9. A bag contains 12 tiles marked 1, 1, 2, 2, ... , 6, 6. A player draws tiles one at a time at random and holds them. If he draws a tile matching a tile he already holds, then he discards both. The game ends if he holds three unmatched tiles or if the bag is emptied. Find the probability that the bag is emptied.

10. ABC is a triangle with ∠C = 90^o. CD is an altitude. BD = 29³, and AC, AD, BC are all integers. Find cos B.

11. Given 94 identical bricks, each 4 x 10 x 19, how many different heights of tower can be built (assuming each brick adds 4, 10 or 19 to the height)?

12. A 24 x 52 field is fenced. An additional 1994 of fencing is available. It is desired to divide the entire field into identical square (fenced) plots. What is the largest number that can be obtained?

13. The equation x¹⁰ + (13x - 1)¹⁰ = 0 has 5 pairs of complex roots a₁, b₁, a₂, b₂, a₃, b₃, a₄, b₄, a₅, b₅. Each pair a_i, b_i are complex conjugates. Find ∑ 1/(a_ib_i).

14. AB and BC are mirrors of equal length. Light strikes BC at C and is reflected to AB. After several reflections it starts to move away from B and emerges again from between the mirrors. How many times is it reflected by AB or BC if ∠b = 1.994^o and ∠a = 19.94^o?

At each reflection the two angles x are equal:

15. ABC is a paper triangle with AB = 36, AC = 72 and ∠B = 90^o. Find the area of the set of points P inside the triangle such that if creases are made by folding (and then unfolding) each of A, B, C to P, then the creases do not overlap.

Answers

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.